Many industrial processes depend on the measurement of material properties as temperature, water contents and material density. A close monitoring of these material properties results often in increased efficiency and improved product quality. Additional benefits are likely to occur, if such measurements can be accomplished fast and in a non-destructive, non-invasive, and non-contacting way with acceptable accuracy.
As an example the determination of the temperature distribution in foodstuff during heating process. Here, an on-line monitoring of the temperature distribution helps to avoid cold spots where bacteria are not eliminated completely or to reduce the overdue heating time spent to ensure complete bacteria elimination. This results in a reduced heating time and reduced energy consumption as well as in an increased throughput of the production line.
Material properties are traditionally measured by some form of destruction (sample separation, peeking) but can often be measured by the analysis of transmitted electromagnetic radiation by evaluating the dielectric response of the material. Measurements using electromagnetic radiation are generally contact-free and non-destructive.
A suitable frequency region of electromagnetic radiation to determine material properties as temperature distribution, water contents and density is the lower microwave region where water absorption is not too large and the wavelength is already short enough to ensure reasonable spatial resolution. The determination of the above material properties is achieved by analysing the dielectric response of the material based on the material's polarisability. Dielectric data of a material sample are typically obtained in analysing the electromagnetic wave's reflection and transmission properties or a combination of both. In order to obtain a distribution of the material properties, a three-dimensional image of the material's dielectric response must be measured. This requires to move the microwave detector setup and the material sample relative to each other.
Prior art instruments make use of either a single measurement frequency or the emission frequency is swept within a frequency interval (FMCW) and the average delay time is calculated from the obtained data.
Prior art that approaches to dielectric imaging, use the transmission of electromagnetic radiation of a single frequency (or a small band) between a multitude of antenna locations or the material sample is shifted and rotated or shifted in two dimensions in order to obtain a spatial resolution. Based on these data the dielectric image is obtained e.g. by the well-known CSI (contrast source iteration) method where the location and strength of polarisation sources are obtained in an iterative process.
Using techniques well-known to a skilled person (i.e. Contrast Source Iteration CSI, as described below) the electromagnetic picture is used to calculate the unknown dielectric functions in the dielectric picture.
Starting from Maxwell's Equations (as described by R. F. Harrington, in the book with the title “Time Harmonic Electromagnetic Fields”, published by Mc. Graw Hill 1961) one assumes that any region where the dielectric function is different from unity, the electromagnetic field creates bound charges due to polarisation. These bound charges are created by the electric field itself and they oscillate with it resulting in an additional current component:j(p)=∈(p)·u(p)
where the current density is j, the electric field is u and the dielectric function of the material is ε and of the background is denoted ∈b·. Assume p and q to be two position vectors in a two dimensional cross section of the measurement gap D is a domain which contains the cross section of the material sample. The vector q denotes the source point of the electromagnetic radiation. Based on that a general relation for the connection between the electric fields in the measurement space is obtained formally by applying the definition of a Green's function for the electric current:
            u      j        ⁡          (      p      )        =            k      2        ⁢                  ∫        D                                      ⁢                        G          ⁡                      (                          p              ,              q                        )                          ·                  j          ⁡                      (            q            )                          ·                  ⅆ                      v            ⁡                          (              q              )                                          
Inserting the above current density relation and splitting the integral yields:
            u      j        ⁡          (      p      )        =                    k        2            ⁢                        ∫          D                                                ⁢                              G            ⁡                          (                              p                ,                q                            )                                ·                      ɛ            b                    ·                      u            ⁡                          (              p              )                                ·                      ⅆ                          v              ⁡                              (                q                )                                                          +                  k        2            ⁢                        ∫          D                                                ⁢                              G            ⁡                          (                              p                ,                q                            )                                ·                      [                                          ɛ                ⁡                                  (                  p                  )                                            -                              ɛ                b                                      ]                    ·                      u            ⁡                          (              p              )                                ·                      ⅆ                          v              ⁡                              (                q                )                                                        
Here the first term denotes the electric field when the dielectric response of the background is present only, the second term stands for the fields generated by polarisation i.e. a dielectric contrast. The fields when only a background is presented are referred to as incident fields uinc. Then the field at an observation point incident from the radiation source is (according to an article by P. M. van den Berg, B. J. Kooj, R. E. Kleinman, with the title “Image Reconstruction from Iswich-Data III”, published in IEEE Antenna and Propagation Magazine, Vol. 41 No. 2 April 1999, p. 27-32):
            u      j        ⁡          (      p      )        =                    u        j        inc            ⁡              (        p        )              +                  k        2            ⁢                          ⁢                        ∫          D                                                ⁢                              G            ⁡                          (                              p                ,                q                            )                                ·                      χ            ⁡                          (              q              )                                ·                                    u              j                        ⁡                          (              q              )                                ·                      ⅆ                          v              ⁡                              (                q                )                                                        
where G denotes the two-dimensional Green's function of the electromagnetic problem
      G    ⁡          (              p        ,        q            )        =            i      4        ⁢                  H        0                  (          1          )                    ⁡              (                  k          ·                                                p              -              q                                                  )            
and the polarisability function χ depends on the dielectric function of the material ε and the background εb· in the following way:
      χ    ⁡          (      p      )        =                    ɛ        ⁡                  (          p          )                    -              ɛ        b                    ɛ      0      
Defining scattered fields f one obtains directly:
                                          F            j                    ⁡                      (            r            )                          =                                            u              j                        ⁡                          (              r              )                                -                                    u              j              inc                        ⁡                          (              r              )                                                                                        ⁢                  =                                    k              2                        ⁢                                          ∫                D                                                                              ⁢                                                G                  ⁡                                      (                                          r                      ,                      q                                        )                                                  ·                                  χ                  ⁡                                      (                    q                    )                                                  ·                                                      u                    j                                    ⁡                                      (                    q                    )                                                  ·                                  ⅆ                                      v                    ⁡                                          (                      q                      )                                                                                                              
From this an integral equation for the scattered electric field at any point r is set up.
            F      j        ⁡          (      r      )        =            i      4        ⁢          k      2        ⁢                  ∫        D                                      ⁢                                    H            0                          (              1              )                                ⁡                      (                          k              ·                                                                r                  -                  q                                                                      )                          ·                  χ          ⁡                      (            q            )                          ·                  [                                                    F                j                            ⁡                              (                q                )                                      +                                          u                j                inc                            ⁡                              (                q                )                                              ]                ·                  ⅆ                      v            ⁡                          (              q              )                                          
This relation is fulfilled exactly when r is equal to the antenna location and the Fi(r) are measured values of the scattered fields for a given wave vector k for a frequency f:
  k  =                    2        ⁢        π            c        ·          f      .      The values of Fi(r) for the points interior to the region D are only fulfilled approximately. So the above relation has to be solved for a set K of k vectors and a set Q of internal points resulting in a [K·Q]×[K·Q] non-linear matrix problem for the fields Fi(r) and the polarisabilities χ(r).
In matrix form the state equation becomes:u=uinc+Gχu
whereas the frequency relation is:F=Gχu
Introducing the contrast source φ=χ·u the above relations become φ=χuinc+χGφ at all Q interior points, for any of the K measurement frequencies F=Gφ at a single antenna location, for any of the K measurement frequencies.
Using the method of conjugated gradients sequences for the contrasts and the contrast sources solving the above problem are obtained.